Integrand size = 6, antiderivative size = 56 \[ \int (x+\sinh (x))^3 \, dx=-\frac {3 x^2}{4}+\frac {x^4}{4}+5 \cosh (x)+3 x^2 \cosh (x)+\frac {\cosh ^3(x)}{3}-6 x \sinh (x)+\frac {3}{2} x \cosh (x) \sinh (x)-\frac {3 \sinh ^2(x)}{4} \]
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Time = 0.05 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.000, Rules used = {6874, 3377, 2718, 3391, 30, 2713} \[ \int (x+\sinh (x))^3 \, dx=\frac {x^4}{4}-\frac {3 x^2}{4}+3 x^2 \cosh (x)-\frac {3 \sinh ^2(x)}{4}-6 x \sinh (x)+\frac {\cosh ^3(x)}{3}+5 \cosh (x)+\frac {3}{2} x \sinh (x) \cosh (x) \]
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Rule 30
Rule 2713
Rule 2718
Rule 3377
Rule 3391
Rule 6874
Rubi steps \begin{align*} \text {integral}& = \int \left (x^3+3 x^2 \sinh (x)+3 x \sinh ^2(x)+\sinh ^3(x)\right ) \, dx \\ & = \frac {x^4}{4}+3 \int x^2 \sinh (x) \, dx+3 \int x \sinh ^2(x) \, dx+\int \sinh ^3(x) \, dx \\ & = \frac {x^4}{4}+3 x^2 \cosh (x)+\frac {3}{2} x \cosh (x) \sinh (x)-\frac {3 \sinh ^2(x)}{4}-\frac {3 \int x \, dx}{2}-6 \int x \cosh (x) \, dx-\text {Subst}\left (\int \left (1-x^2\right ) \, dx,x,\cosh (x)\right ) \\ & = -\frac {3 x^2}{4}+\frac {x^4}{4}-\cosh (x)+3 x^2 \cosh (x)+\frac {\cosh ^3(x)}{3}-6 x \sinh (x)+\frac {3}{2} x \cosh (x) \sinh (x)-\frac {3 \sinh ^2(x)}{4}+6 \int \sinh (x) \, dx \\ & = -\frac {3 x^2}{4}+\frac {x^4}{4}+5 \cosh (x)+3 x^2 \cosh (x)+\frac {\cosh ^3(x)}{3}-6 x \sinh (x)+\frac {3}{2} x \cosh (x) \sinh (x)-\frac {3 \sinh ^2(x)}{4} \\ \end{align*}
Time = 0.06 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.86 \[ \int (x+\sinh (x))^3 \, dx=\frac {1}{24} \left (18 \left (7+4 x^2\right ) \cosh (x)-9 \cosh (2 x)+2 \cosh (3 x)+6 x \left (-3 x+x^3-24 \sinh (x)+3 \sinh (2 x)\right )\right ) \]
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Time = 0.42 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.86
method | result | size |
parallelrisch | \(-\frac {47}{24}+\frac {x^{4}}{4}-\frac {3 x^{2}}{4}+3 x^{2} \cosh \left (x \right )-6 x \sinh \left (x \right )+\frac {3 x \sinh \left (2 x \right )}{4}+\frac {\cosh \left (3 x \right )}{12}+\frac {21 \cosh \left (x \right )}{4}-\frac {3 \cosh \left (2 x \right )}{8}\) | \(48\) |
default | \(\left (-\frac {2}{3}+\frac {\sinh \left (x \right )^{2}}{3}\right ) \cosh \left (x \right )+\frac {3 x \cosh \left (x \right ) \sinh \left (x \right )}{2}-\frac {3 x^{2}}{4}-\frac {3 \cosh \left (x \right )^{2}}{4}+3 x^{2} \cosh \left (x \right )-6 x \sinh \left (x \right )+6 \cosh \left (x \right )+\frac {x^{4}}{4}\) | \(52\) |
parts | \(\left (-\frac {2}{3}+\frac {\sinh \left (x \right )^{2}}{3}\right ) \cosh \left (x \right )+\frac {3 x \cosh \left (x \right ) \sinh \left (x \right )}{2}-\frac {3 x^{2}}{4}-\frac {3 \cosh \left (x \right )^{2}}{4}+3 x^{2} \cosh \left (x \right )-6 x \sinh \left (x \right )+6 \cosh \left (x \right )+\frac {x^{4}}{4}\) | \(52\) |
risch | \(\frac {x^{4}}{4}-\frac {3 x^{2}}{4}+\frac {9}{16}+\frac {{\mathrm e}^{3 x}}{24}+\left (-\frac {3}{16}+\frac {3 x}{8}\right ) {\mathrm e}^{2 x}+\left (\frac {21}{8}-3 x +\frac {3}{2} x^{2}\right ) {\mathrm e}^{x}+\left (\frac {21}{8}+3 x +\frac {3}{2} x^{2}\right ) {\mathrm e}^{-x}+\left (-\frac {3}{16}-\frac {3 x}{8}\right ) {\mathrm e}^{-2 x}+\frac {{\mathrm e}^{-3 x}}{24}\) | \(73\) |
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Time = 0.28 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.04 \[ \int (x+\sinh (x))^3 \, dx=\frac {1}{4} \, x^{4} + \frac {1}{12} \, \cosh \left (x\right )^{3} + \frac {1}{8} \, {\left (2 \, \cosh \left (x\right ) - 3\right )} \sinh \left (x\right )^{2} - \frac {3}{4} \, x^{2} + \frac {3}{4} \, {\left (4 \, x^{2} + 7\right )} \cosh \left (x\right ) - \frac {3}{8} \, \cosh \left (x\right )^{2} + \frac {3}{2} \, {\left (x \cosh \left (x\right ) - 4 \, x\right )} \sinh \left (x\right ) \]
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Time = 0.09 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.52 \[ \int (x+\sinh (x))^3 \, dx=\frac {x^{4}}{4} + \frac {3 x^{2} \sinh ^{2}{\left (x \right )}}{4} - \frac {3 x^{2} \cosh ^{2}{\left (x \right )}}{4} + 3 x^{2} \cosh {\left (x \right )} + \frac {3 x \sinh {\left (x \right )} \cosh {\left (x \right )}}{2} - 6 x \sinh {\left (x \right )} + \sinh ^{2}{\left (x \right )} \cosh {\left (x \right )} - \frac {2 \cosh ^{3}{\left (x \right )}}{3} - \frac {3 \cosh ^{2}{\left (x \right )}}{4} + 6 \cosh {\left (x \right )} \]
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Time = 0.23 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.45 \[ \int (x+\sinh (x))^3 \, dx=\frac {1}{4} \, x^{4} - \frac {3}{4} \, x^{2} + \frac {3}{16} \, {\left (2 \, x - 1\right )} e^{\left (2 \, x\right )} + \frac {3}{2} \, {\left (x^{2} + 2 \, x + 2\right )} e^{\left (-x\right )} - \frac {3}{16} \, {\left (2 \, x + 1\right )} e^{\left (-2 \, x\right )} + \frac {3}{2} \, {\left (x^{2} - 2 \, x + 2\right )} e^{x} + \frac {1}{24} \, e^{\left (3 \, x\right )} - \frac {3}{8} \, e^{\left (-x\right )} + \frac {1}{24} \, e^{\left (-3 \, x\right )} - \frac {3}{8} \, e^{x} \]
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Time = 0.25 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.34 \[ \int (x+\sinh (x))^3 \, dx=\frac {1}{4} \, x^{4} - \frac {3}{4} \, x^{2} + \frac {3}{16} \, {\left (2 \, x - 1\right )} e^{\left (2 \, x\right )} + \frac {3}{8} \, {\left (4 \, x^{2} + 8 \, x + 7\right )} e^{\left (-x\right )} - \frac {3}{16} \, {\left (2 \, x + 1\right )} e^{\left (-2 \, x\right )} + \frac {3}{8} \, {\left (4 \, x^{2} - 8 \, x + 7\right )} e^{x} + \frac {1}{24} \, e^{\left (3 \, x\right )} + \frac {1}{24} \, e^{\left (-3 \, x\right )} \]
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Time = 0.08 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.82 \[ \int (x+\sinh (x))^3 \, dx=5\,\mathrm {cosh}\left (x\right )+3\,x^2\,\mathrm {cosh}\left (x\right )-\frac {3\,{\mathrm {cosh}\left (x\right )}^2}{4}+\frac {{\mathrm {cosh}\left (x\right )}^3}{3}-6\,x\,\mathrm {sinh}\left (x\right )-\frac {3\,x^2}{4}+\frac {x^4}{4}+\frac {3\,x\,\mathrm {cosh}\left (x\right )\,\mathrm {sinh}\left (x\right )}{2} \]
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